The Principle of Dynamic Inequality Equilibrium : A Unified Framework for Diagnosing the Emergence of Scale-Free Structures

Many mechanisms generate heavy-tailed and scale-free distributions: random multiplica-tive growth with reinjection (Kesten), stabilized geometric Brownian motion with boundaries,preferential attachment with entry, and various critical phenomena. Static tail fitting is rarelyidentifying because distinct dynamics can produce similar apparent tail indices. This paperdoes not propose a new generative theorem for power laws. Instead it develops a compara-tive, falsifiable framework—Dynamic Inequality Equilibrium —that uses inequality dynamics(Lorenz/convex order) to constrain broad mechanism classes.Conceptually, Class A is the inequality-concentrating drive and Class B is the super-categoryof stabilizers that stabilize Class A. For rigor and diagnostics, Class B is subdivided into Type I(affine reinjection: a global baseline in levels) and Type II (boundary/entry stabilization: floors,reflection, resets, and entry flux). The paper leverages classical foundations from majorizationtheory and iterated random functions to derive disciplined diagnostic implications, including astationary-mean identity that yields a sharp, testable constraint for Type I stabilization. Wevalidate the diagnostic bundle by simulation (showing separation even when Hill tail indices aresimilar) and illustrate it on US place populations and a firm-size-bin control series.